A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into continuous partial (PDE). We show that can be performed for both linear and nonlinear systems, including multidimensional, space-varying, time-varying systems. When applied to large-scale network, the provides PDE describing evolution of continuous-state approximation respects spatial structure original ODE. Our method is illustrated by multiple examples, transport equations, Kuramoto heat diffusion equations. As main example, perform Newtonian system interacting particles obtain Euler equations compressible fluids, thereby providing an solution Hilbert’s sixth problem. Finally, leverage our derivation solve control problem multiagent designing algorithm robot formation based on its approximation.